Well. The city does not actually produce food. Stores get their produce from wholesalers. This is where the "city stores" will get them as well. Only now, regular stores know the city will undercut them in price using taxpayer money. So any location next to a city store will become poison. And soon all you will have (for some people) are the city stores.
Of course, with prices much cheaper than in other parts, you will likely get crowding at these stores. Leading to the infamous bread lines of the Soviet Union or Venezuela or what have you. Basically, if something is cheaper than the market price, you get shortages.
Ironically, the city sets prices based on a relative level to "normal stores". So it's not as if this is some "good capitalist competition". It's not as if it's a "temporary measure until catastrophe A is over".
There is put simply no way this experiment ends in a way that is positive for New Yorkers. And yes, this is socialism and there are very logical, rational economic reasons for why it always fail. This will be no different.